J Sci Comput (2012) 52:638–655
DOI 10.1007/s10915-011-9564-5
Fourier Type Error Analysis of the Direct Discontinuous
Galerkin Method and Its Variations for Diffusion
Equations
Mengping Zhang · Jue Yan
Received: 28 September 2011 / Revised: 21 November 2011 / Accepted: 22 November 2011 /
Published online: 7 December 2011
© Springer Science+Business Media, LLC 2011
Abstract In this paper we present Fourier type error analysis on the recent four discontinuous Galerkin methods for diffusion equations, namely the direct discontinuous Galerkin
(DDG) method (Liu and Yan in SIAM J. Numer. Anal. 47(1):475–698, 2009); the DDG
method with interface corrections (Liu and Yan in Commun. Comput. Phys. 8(3):541–564,
2010); and the DDG method with symmetric structure (Vidden and Yan in SIAM J. Numer. Anal., 2011); and a DG method with nonsymmetric structure (Yan, A discontinuous
Galerkin method for nonlinear diffusion problems with nonsymmetric structure, 2011). The
Fourier type L2 error analysis demonstrates the optimal convergence of the four DG methods
with suitable numerical fluxes. The theoretical predicted errors agree well with the numerical results.
Keywords Discontinuous Galerkin method · Diffusion equation · Stability · Consistency ·
Convergence · Supraconvergence
1 Introduction
The focus of this paper is to perform Fourier type error analysis on four discontinuous
Galerkin methods for diffusion problems. We have (1) the direct discontinuous Galerkin
(DDG) method [16]; (2) the DDG method with interface corrections [17]; (3) a variation of
the DDG method with symmetric structure [26]; and (4) a discontinuous Galerkin method
The research of M. Zhang is supported by NSFC grant 11071234 and 91024025.
The research of J. Yan is supported by NSF grant DMS-0915247.
M. Zhang
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026,
P.R. China
e-mail: mpzhang@ustc.edu.cn
J. Yan ()
Department of Mathematics, Iowa State University, Ames, IA 50011, USA
e-mail: jyan@iastate.edu
J Sci Comput (2012) 52:638–655
639
with nonsymmetric structure [30]. In this paper, for the simplicity of presentation we present
the four discontinuous Galerkin methods for the diffusion equations on the simple one dimensional linear heat equation
Ut − Uxx = 0,
for x ∈ [0, 2π]
(1.1)
with suitable given boundary conditions and initial condition U (x, 0) = sin(αx). We would
like to point out, however, that the four discontinuous Galerkin methods are actually designed and can be analyzed for much more general multidimensional nonlinear convection
diffusion equations, see, e.g. [17]. The points we would like to make in this paper can be
represented well by the simple case (1.1).
The discontinuous Galerkin (DG) method is a finite element method using a completely
discontinuous piecewise function space for the numerical solution and the test function. One
main feature of the DG method is the flexibility afforded by local approximation spaces
combined with the suitable design of numerical fluxes crossing cell interfaces. The application to hyperbolic problems has been quite successful since it was originally introduced
by Reed and Hill [19] in 1973 for neutron transport equations. A major development of the
DG method for nonlinear hyperbolic conservation laws is carried out by Cockburn, Shu, and
collaborators. We refer to [10, 11, 13] for reviews and further references.
However, the application of the DG method to diffusion problems has been a challenging
task because of the subtle difficulty in defining appropriate numerical fluxes for diffusion
terms, see e.g. [21]. There have been several DG methods suggested in literature to solve
the problem. One class is the interior penalty (IP) methods, dates back to 1982 by Arnold
in [1] (also by Baker in [3] and Wheeler in [27]), the Baumann and Oden method [5, 18],
the NIPG method [20] and the IIPG method [14]. Another class is the local discontinuous
Galerkin (LDG) method introduced in [12] by Cockburn and Shu (originally studied by
Bassi and Rebay in [4] for compressible Navier-Stokes equations). Here we refer to the
unified analysis paper [2] in 2002 for the review of different DG methods for diffusion
equation. More recent works for diffusion with DG methods are those by Van Leer and
Nomura in [24], Gassner et al. in [15], Cheng and Shu in [7], and Brenner et al. in [6].
Recently in [16] we introduce a discontinuous Galerkin method to solve diffusion equations. The scheme is based on the direct weak formulation, thus we name it the direct discontinuous Galerkin (DDG) method. We propose a general numerical flux formula for the
solution derivative ux at the cell interface, which involves the average ux and even order
derivative jumps [∂x2m u] (m = 0, 1, . . . , [ k2 ]) across the cell interfaces. We then introduce a
concept of admissibility to identify suitable numerical fluxes to further obtain stability and
energy norm error estimate. However, numerical experiments in [16] show that measured in
L2 norm the scheme accuracy is sensitive to the coefficients chosen in the numerical flux
formula, and for higher order p k (k ≥ 4) polynomial approximations it is difficult to identify suitable coefficients in the numerical flux formula to obtain optimal (k + 1)th order of
accuracy.
In [17], we introduce a refined version of the DDG method by adding extra interface
correction terms such that a simpler numerical flux formula can be used and numerically
optimal accuracy under L2 norm can be achieved for all p k polynomials. The interface
terms are added due to the observation that the test function is also discontinuous at the cell
interface and the derivative of the test function does contribute to the interface flux when
higher order approximations are used. Another feature of the refined DDG method is that
the scheme is not sensitive to the coefficients in the numerical flux formula. Numerical tests
show a large class of admissible numerical fluxes can lead to optimal order of accuracy.
640
J Sci Comput (2012) 52:638–655
More recently in [26], we further explore the DDG method by introducing a numerical
flux for the test function derivative vx and adding more interface terms in the scheme formulation, and obtain a DDG scheme with symmetric structure. The symmetric structure helps
us prove the optimal (k + 1)th order L2 (L2 ) error estimate with any p k polynomial approximations. Detailed admissibility analysis confirms that there exists a large class of admissible
numerical fluxes, basically any coefficients pair that falls in a parabolic region is admissible
(see [26] or details in [25]).
When adding the test function derivative vx interface terms in a negative sense, we end
up with a discontinuous Galerkin method with nonsymmetric structure in [30]. This nonsymmetric DG method is closely related to the Baumann-Oden method [5] and the NIPG
method [20]. Essentially the scheme degenerates to the Baumann-Oden or the NIPG scheme
if the second derivative jump term [uxx ] is not included in the numerical flux formula for ux
(same as [vxx ] in vx ). On the other hand, numerically we observe our nonsymmetric scheme
can lead to optimal (k + 1)th order convergence with any p k polynomial approximations.
As we know, only sub-optimal kth order of accuracy can be obtained for the Baumann-Oden
method and the NIPG method with p k (k = even) polynomial approximations. In a word,
the optimal accuracy is recovered with our nonsymmetric scheme [30].
Notice that it is difficult to obtain L2 error analysis for the DDG method [16], the DDG
method with interface corrections [17], and the nonsymmetric DG method [30]. On the
other hand, Fourier analysis is a new technique to study the stability and error estimates
for the discontinuous Galerkin method and other related schemes, especially in some cases
where standard finite element technique can not be applied. Numerically we observe the
DDG method [16] is sensitive to the coefficients in the numerical flux formula to obtain
optimal convergence with even-th order polynomial approximations, thus in this paper we
focus on the p 2 quadratic case and use the Fourier analysis technique to analyze the error
behavior of the four DG methods. The Fourier analysis does have several advantages over
the standard finite element techniques. It can be used to analyze some of the “bad” schemes
[31]; it can be used for stability analysis for some of the non-standard methods such as the
spectral volume (SV) method [32], which belongs to the class of Petrov-Galerkin methods
and cannot be easily amended to the standard finite element analysis framework; it can
provide quantitative error comparisons among different schemes [28, 29]; and it can be used
to prove superconvergence and time evolution of errors for the DG method [8, 9].
This rest of the paper is organized as follows. In Sect. 2, we describe the four DG methods
for the model heat equation. In Sect. 3, we write out in detail the Fourier technique to analyze
the errors of the four DG methods, and show the numerical errors match well with the
analytical predictions. Finally, some concluding remarks are given in Sect. 4.
2 Direct Discontinuous Galerkin Method and Its Variations
Again, in this paper we focus on the model heat equation
Ut − Uxx = 0,
for x ∈ [0, 2π]
with initial condition u(x, 0) = sin(αx) and suitable given boundary condition.
Let us denote Ij =[xj − 1 , xj + 1 ], j = 1, . . . , N , as a mesh for [0, 2π ], where x 1 = 0 and
2
2
2
xN+ 1 = 2π . We denote the center of each cell by xj = 12 (xj − 1 + xj + 1 ) and the size of each
2
2
2
cell by xj = xj + 1 − xj − 1 . The cells do not need to be uniform for the numerical methods,
2
2
but for simplicity of analysis we will consider only uniform meshes in this paper and will
J Sci Comput (2012) 52:638–655
641
denote the uniform mesh size by x. The numerical solution u and the test function v are
piecewise polynomials of degree k. In a word, for any time t ∈ [0, T ], u ∈ Vx , where
Vx := {v ∈ L2 (0, 2π) : v|Ij ∈ P k (Ij ), j = 1, . . . , N },
and P k (Ij ) denotes the space of polynomials in Ij with degree at most k. In this paper we
use lower letters to represent numerical solutions and test functions. We are now ready to
formulate the DG schemes.
2.1 Direct Discontinuous Galerkin Method
We multiply heat equation (1.1) by an arbitrary test function v(x) ∈ Vx , integrate over the
cell Ij , have the integration by parts, and formally we obtain,
Ij
where
j + 1 2
ut vdx − ux v 1 +
ux vx dx = 0,
j− 2
(2.1)
Ij
j + 1
2
ux v 1 := (ux )j + 1 vj−+ 1 − (ux )j − 1 vj+− 1 .
j− 2
2
2
2
2
This is the starting point to design the direct discontinuous Galerkin method. Note here
and below we adopt the following notations:
u± = u(x ± 0, t),
[u] = u+ − u− ,
u=
u+ + u−
.
2
Motivated by the solution derivative trace formula of the heat equation with discontinuous
initial data, in [16] we introduce a numerical flux formula at the cell interface xj ±1/2 by the
following form,
ux = β0
[u]
+ ux + β1 x[uxx ] + β2 (x)3 [uxxxx ] + · · · .
x
(2.2)
The numerical flux ux which approximates ux at the discontinuous cell interface xj ±1/2
thus involves the average ux and the jumps of even order derivatives of u. The coefficients
β0 , β1 , . . . are chosen to ensure the stability and convergence of the method. The numerical flux ux thus defined is conservative and consistent to the solution derivative. We then
introduce a concept of admissibility for the numerical fluxes. The admissibility condition
serves as a criterion for selecting suitable numerical fluxes (βi ’s) to guarantee nonlinear stability of the DDG method and the corresponding convergence of the method. Indeed in the
linear case, for the error in a parabolic energy norm L∞ (0, T ; L2 ) ∩ L2 (0, T ; H 1 ), convergence rate of order (x)k is obtained for p k polynomial approximations. Numerically we
observe that the scheme is sensitive to the choice of βi ’s to obtain optimal (k + 1)th order
of accuracy under L2 norm, especially for the case k = even. For p 2 quadratic polynomial
approximation, only the numerical flux
ux = β0
x
[u]
+ ux +
[uxx ]
x
12
1
gives optimal 3rd order of accuracy, all other admissible numerical fluxes with
with β1 = 12
1
β1 = 12 degenerate the scheme to a 2nd order. On the other hand, the scheme accuracy is not
642
J Sci Comput (2012) 52:638–655
sensitive to the choice of β0 . In the following Sect. 3.1 the Fourier type error analysis con1
gives the optimal 3rd order of accuracy. As
firms these numerical results that only β1 = 12
for higher order polynomial approximations with even k and k ≥ 4, it remains an unsettled
issue on how to find suitable βi ’s to ensure optimal (k + 1)th order of accuracy.
2.2 Direct Discontinuous Galerkin Method with Interface Corrections
In [17], we add interface correction terms in the DDG scheme formulation and obtain the
following refined DDG method,
Ij
j + 1 1
1
2
+
ut vdx − ux v 1 +
ux vx dx + [u](vx )−
j +1/2 + [u](vx )j −1/2 = 0.
j− 2
2
2
Ij
(2.3)
The Numerical flux is taken as
ux = β0
[u]
+ ux + β1 x[uxx ],
x
(2.4)
where the second derivative jump term is still included and higher order (k ≥ 4) derivatives
jump terms are dropped. Extra interface terms are added due to the observation that the test
function v is also discontinuous at the cell interfaces, the slope of the test function will contribute at interfaces whenever [u] is non-zero. The refined DDG method with such numerical
fluxes enjoys the optimal (k + 1)th order of accuracy for all p k polynomial approximations.
If we take β1 = 0 in (2.4), the numerical flux reduces to
ux = β0
[u]
+ ux ,
x
(2.5)
in which case the scheme reduces to the classical symmetric Interior Penalty (IP) method
[1]. It is well known that the penalty parameter (β0 in (2.5)) depends on k the order of the
approximate polynomial, and needs to be large enough to stabilize the scheme.
With β1 = 0 in (2.4), numerically we observe that (k + 1)th order of accuracy is obtained
for p k polynomials with a fixed β0 . For instance, optimal accuracy is obtained for all p k up
1
) in (2.4). Thus the second derivative jump term [uxx ]
to k = 9 when taking (β0 , β1 ) = (2, 12
is important, it indeed provides leverage to compensate the β0 term. Moreover, there exists a
large class of β0 and β1 that lead to optimal order of accuracy with p k polynomial approximations. In the following Sect. 3.2, the Fourier type error analysis results are consistent with
these numerical tests.
2.3 Direct Discontinuous Galerkin Method with Symmetric/Nonsymmetric Structure
We see the DDG method (2.1)–(2.2) and the DDG method with interface corrections (2.3)–
(2.4) are schemes lack of symmetric structures, and thus are not adjoint-consistent. To further study the DDG method, we introduce a numerical flux term vx for the test function v
and add it into the original DDG scheme (2.1). This test function numerical flux vx shares
similar format to the solution numerical flux ux given above. With more interface terms, now
we obtain the following DG methods with symmetric (σ = +1) or nonsymmetric structures
(σ = −1),
Ij
j + 1 2
ut vdx − ux v 1 +
ux vx dx + σ (
vx [u]j +1/2 + vx [u]j −1/2 ) = 0,
j− 2
Ij
(2.6)
J Sci Comput (2012) 52:638–655
643
where the numerical flux is defined as
ux = β0u
[u]
+ ux + β1 x[uxx ],
x
(2.7)
[v]
+ vx + β1 x[vxx ].
vx = β0v
x
Notice that the test function v is taken to be none zero only inside the cell Ij , thus at the
cell interface xj ±1/2 only one side contributes to the computation of vx . Now, summing the
scheme (2.6) over all computational cells Ij (here we assume the periodic boundary condition for simplicity of explanation), we obtain the primal weak formulation as the following,
2π
ut vdx + B(u, v) = 0,
(2.8)
0
with the bilinear form defined as,
B(u, v) =
N j =1
ux vx dx +
Ij
N
N
(ux [v])j +1/2 + σ
([u]
vx )j +1/2 .
j =1
(2.9)
j =1
With the numerical flux formula of ux and vx as defined in (2.7), it can be seen that the
bilinear form B(u, v) has an obvious symmetry for σ = +1 case. That is, B(u, v) = B(v, u).
In [26], this symmetric structure helps us prove the optimal (k + 1)th order of accuracy in
the L2 (L2 ) sense for any p k polynomial approximations. The admissibility analysis in [26]
shows us that any β0 = β0u + β0v and β1 coefficients that satisfy the following quadratic
form inequality lead to an admissible numerical flux (2.7), and guarantees the optimal convergence of the symmetric DDG scheme,
2 2
2 2
k (k − 1)2
k (k − 1)
k2
β0u + β0v ≥ 1 + 8 β12
− β1
+
,
3
2
4
for k ≥ 1.
For the purpose of this paper where we mainly consider piecewise quadratic p 2 approximations, here we choose β0u = β0v = 1 and β1 = 1/12 in the following Sect. 3.3 to carry out
the Fourier analysis of the symmetric DDG method.
Taking σ = −1 in (2.6), we end up having a new DG method with nonsymmetric structure in [30]. This nonsymmetric DG method is closely related to the Baumann-Oden method
[5] and the NIPG method [20]. The essential difference is that the second derivative jump
term [uxx ] and [vxx ] are included in the numerical flux formula (2.7). When taking v = u
in (2.8) to proceed the stability analysis, we obtain a stability result that is the same as the
Baumann-Oden method (with β0u = β0v ) and the NIPG method (with β0u > β0v ) as below,
1 d
2 dt
2π
u dx +
2
0
N j =1
(ux )2 dx + (β0u − β0v )
Ij
N
[u]2j +1/2
j =1
x
= 0.
(2.10)
Even our nonsymmetric DG method shares the same stability with the Baumann-Oden
method and the NIPG method, the numerical performance is quite different. Numerically
we obtain (k + 1)th order of accuracy with any p k polynomial approximations for our nonsymmetric DG method [30]. As is well known that both the Baumann-Oden method and the
NIPG method lose one order for p k polynomial approximations when k is even. With the
644
J Sci Comput (2012) 52:638–655
second derivative jump term included in the numerical flux we see the optimal convergence
is recovered. In the following Sect. 3.3, we take β0u = 1 and β0v = 1 coupled with β1 = 1/12
in (2.7) to perform Fourier analysis of the nonsymmetric DG method, the Fourier type error
analysis further confirms the optimal convergence of the method.
Up to now, we have taken the method of lines approach and have left time variable
t continuous. For time discretization, the explicit third order TVD Runge-Kutta method
[22, 23] was used in order to match the accuracy in space. CFL conditions are taken as the
standard ones for DG methods, see [10].
3 Fourier Analysis for the L2 Errors
In this section, we will rewrite the four discontinuous Galerkin methods as finite difference
schemes under the assumption of periodic boundary condition and uniform mesh, then perform Fourier type error analysis on the four schemes and compare the error estimates with
the numerical results. For the L2 error we show that optimal order of convergence can be
obtained for all the four DG methods with suitable numerical fluxes.
We use the direct discontinuous Galerkin scheme (2.1) to demonstrate the procedure of
the Fourier analysis. After picking a local basis for the space Vx and inverting a local
(k + 1) × (k + 1) mass matrix (which could be done by hand), the DDG scheme (2.1) can
be written as
1
d
(A
uj −1 + B uj + C uj +1 )
uj =
dt
x 2
(3.1)
where uj is a small vector of length k + 1 containing the coefficients of the solution u in
the local basis inside cell Ij , and A, B and C are (k + 1) × (k + 1) constant matrices. If we
choose the degrees of freedom for the kth degree polynomial inside the cell Ij as the point
values of the solution, denoted by
uj +
2i−k
2(k+1)
i = 0, . . . , k,
,
at the k + 1 equally spaced points
2i − k
1
j− +
x,
2 2(k + 1)
i = 0, . . . , k,
then the DDG scheme written in terms of these degrees of freedom becomes a finite difference scheme on a globally uniform mesh (with a mesh size x/(k + 1)); however they
are not standard finite difference schemes because each point in the group of k + 1 points
belonging to the cell Ij obeys a different form of finite difference scheme. Let us discuss the
procedure in detail using the piecewise quadratic case k = 2. The degrees of freedom are
now the point values at the 3N uniformly spaced points
uj − 1 , uj , uj + 1 ,
3
3
j = 1, . . . , N.
The solution inside the cell Ij is then represented by
u(x) = uj − 1 φj − 1 (x) + uj φj (x) + uj + 1 φj + 1 (x),
3
3
3
3
J Sci Comput (2012) 52:638–655
645
where φj − 1 (x), φj (x), φj + 1 (x) are the Lagrange interpolation polynomials at points xj − 1 ,
3
3
3
xj , xj + 1 . Now we obtain the finite difference representation of the DDG method (2.1)–(2.2)
3
as in (3.1) with solution vector defined as
⎞
⎛
uj − 1
3
⎟
⎜
(3.2)
uj = ⎝ uj ⎠
uj + 1
3
and scheme matrices defined as
⎛ 23
(−4 + β0 + 24β1 )
32
⎜
A = ⎝ −9
(−4 + β0 + 24β1 )
32
−1
(−4 + β0 + 24β1 )
32
⎛ −3
(6 + 19β0 + 88β1 )
16
⎜ 9
B = ⎝ 16
(14 + 3β0 + 24β1 )
3
(10 − 3β0 + 88β1 )
16
⎛ 1
(8 − 5β0 − 24β1 )
32
⎜ 9
C = ⎝ 32
(8 − 5β0 − 24β1 )
−23
(8 − 5β0 − 24β1 )
32
−24
(−18 + 5β0 + 72β1 )
48
3
(−18
+ 5β0 + 72β1 )
16
1
(−18 + 5β0 + 72β1 )
48
−1
(18 − 55β0
24
−3
(42 + 5β0
8
−1
(18 − 55β0
24
− 814β1 )
+ 72β1 )
− 814β1 )
1
(−18 + 5β0 + 72β1 )
48
3
(−18 + 5β0 + 72β1 )
16
−1
(−18 + 5β0 + 72β1 )
48
⎞
23
(−8 + 5β0 + 24β1 )
32
⎟
−9
(−8 + 5β0 + 24β1 ) ⎠
32
−1
(−8 + 5β0 + 24β1 )
32
⎞
3
(10 − 3β0 − 88β1 )
16
⎟
9
(14 + 3β0 + 24β1 ) ⎠
16
−3
(6 + 19β0 + 88β1 )
16
(3.3)
⎞
1
(4 − β0 − 24β1 )
32
⎟
9
(4 − β0 − 24β1 ) ⎠ .
32
−23
(4 − β0 − 24β1 )
32
We perform the following standard Fourier analysis as on a finite difference method.
The Fourier analysis depends on the assumption of uniform mesh and periodic boundary
condition. We make an ansatz of the solution with the form
⎞ ⎛
⎞
⎛
ûj − 1 (t)
uj − 1 (t)
3
3
⎟ ⎜
⎟ iαx
⎜
(3.4)
⎝ uj (t) ⎠ = ⎝ ûj (t) ⎠ e j
uj + 1 (t)
ûj + 1 (t)
3
3
and substitute the ansatz (3.4) in the DDG scheme (3.1) to find the evolution equation for
the coefficient vector as
⎞
⎛
⎞
⎛
ûj − 1 (t)
û 1 (t)
3
d ⎜ j− 3 ⎟
⎜
⎟
(3.5)
⎝ ûj (t) ⎠ = G(α, x) ⎝ ûj (t) ⎠
dt û (t)
(t)
û
1
1
j+
j+
3
3
where the amplification matrix G(α, x) is given by
G(α, x) =
1 −iαx
+ B + Ceiαx ,
Ae
2
x
(3.6)
and matrices A, B, C are from (3.3), which is the reformulation of the DDG scheme (2.1)
with Lagrange interpolation polynomials as basis functions. Now the error estimates can be
performed based on the matrix G(α, x). The general solution of the ODE (3.5) is given by
⎞
⎛
ûj − 1 (t)
3
⎟
⎜
λ t
λ t
λ t
(3.7)
⎝ ûj (t) ⎠ = a1 e 1 V1 + a2 e 2 V2 , +a3 e 3 V3
ûj + 1 (t)
3
646
J Sci Comput (2012) 52:638–655
where λ1 , λ2 , λ3 , and V1 , V2 , V3 are the eigenvalues and the corresponding eigenvectors of
the amplification matrix G(α, x) respectively.
To fit the given initial condition u(x, 0) = sin(αx), we have
uj − 1 (0) = e
iαx
3
j − 13
,
uj (0) = eiαxj ,
uj + 1 (0) = e
iαx
j + 13
3
(3.8)
whose imaginary part is our initial condition for the heat equation. Thus we require, at t = 0,
⎛
⎞ ⎛ −αx ⎞
ûj − 1 (0)
ei 3
3
⎜
⎟ ⎜
⎟ iαx
⎝ ûj (0) ⎠ = ⎝ 1 ⎠ e j .
αx
ûj + 1 (0)
ei 3
3
This initial condition fitting determines the coefficients a1 , a2 , and a3 in the general solution formula (3.7). Thus we can explicitly write out the solution of the DDG scheme (2.1).
Through a simple Taylor expansion, we can check the imaginary part of uj ± 1 (t), uj (t) and
3
find out the leading order in the error. In the following sections, we will perform the Fourier
analysis to the four discontinuous Galerkin methods for the model heat equation with p2
quadratic polynomial approximations. The predicted errors through the above Fourier analysis match well with the numerical solution errors.
3.1 DDG Method
In this section, we perform Fourier analysis on the DDG method (2.1) with numerical flux
taken in the form of
[u]
+ ux + β1 x[uxx ].
ux = β0
x
1
1
); II: (β0 , β1 ) = (3, 12
); and III:
We carry out three cases studies with I: (β0 , β1 ) = ( 76 , 12
1
(β0 , β1 ) = (3, 4 ). We pick three groups of (β0 , β1 ) in the numerical flux formula to show
that the DDG scheme accuracy is sensitive to the β1 coefficient, but not to the β0 coefficient.
1
and the DDG scheme
It turns out that 3rd order of accuracy is obtained only with β1 = 12
1
degenerates to 2nd order with β1 = 12 .
Case I: β0 =
7
6
and β1 =
1
12
In this case, the numerical flux is given as
ux =
7 [u]
x
+ ūx +
[uxx ].
6 x
12
Through the Fourier analysis, we obtain the DDG solution in cell Ij as
2
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) +
5
2
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
648
(3.9)
and
2
Im{uj (t)} = e−α t sin(αxj ) +
−23 + 132t
2
x 4 e−α t sin(αxj ) + O(x 6 ).
60480
(3.10)
The result for Im{uj + 1 (t)} is similar to Im{uj − 1 (t)}. With the initial condition u(x, 0) =
3
3
2
sin(αx), we know the exact solution of the heat equation is u(x, t) = e−α t sin(αx). We see
J Sci Comput (2012) 52:638–655
647
Table 1 DDG scheme (2.1) with Case I numerical flux. L2 and L∞ errors for u, measured at the point xj − 1
3
of the cells
x
Numerical solutions
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
1.08E-05
–
1.53E-05
–
2.11E-05
–
2.98E-05
–
2π/80
2.49E-06
2.12
3.52E-06
2.12
2.64E-06
3.00
3.73E-06
3.00
2π/160
3.30E-07
2.92
4.67E-07
2.92
3.30E-07
3.00
4.67E-07
3.00
2π/320
4.13E-08
3.00
5.83E-08
3.00
4.13E-08
3.00
5.83E-08
3.00
2π/640
5.16E-09
3.00
7.29E-09
3.00
5.16E-09
3.00
7.29E-09
3.00
Order
Table 2 DDG scheme (2.1) with Case I numerical flux. L2 and L∞ errors for u, measured at the center xj
of the cells
x
Numerical solutions
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
2.48E-07
–
3.51E-07
–
1.62E-07
–
2.30E-07
–
2π/80
1.09E-08
4.51
1.54E-08
4.51
1.02E-08
4.00
1.44E-08
4.00
2π/160
6.34E-10
4.11
8.97E-10
4.11
6.34E-10
4.00
8.97E-10
4.00
2π/320
3.96E-11
4.00
5.61E-11
4.00
3.96E-11
4.00
5.61E-11
4.00
2π/640
2.48E-12
4.00
3.50E-12
4.00
2.48E-12
4.00
3.50E-12
4.00
L∞ error
Order
Order
Table 3 DDG scheme (2.1) with numerical flux of Case I and Case II
x
1
Case I: β0 = 76 , β1 = 12
1
Case II: β0 = 3, β1 = 12
L2 error
Order
L∞ error
Order
L2 error
Order
2π/40
5.45E-05
–
2.19E-05
–
7.55E-05
–
2.70E-05
–
2π/80
1.06E-05
2.36
3.69E-06
2.57
1.12E-05
2.75
3.87E-06
2.80
2π/160
1.41E-06
2.92
4.84E-07
2.93
1.41E-06
3.00
4.84E-07
3.00
2π/320
1.76E-07
3.00
6.05E-08
3.00
1.76E-07
3.00
6.05E-08
3.00
2π/640
2.20E-08
3.00
7.56E-09
3.00
2.20E-08
3.00
7.56E-09
3.00
the leading errors for points xj − 1 and xj + 1 are 3rd order and is 4th order for point xj . We
3
3
compare the numerical solution errors and the analytical errors (the leading terms in (3.10)
and (3.9)) with α = 0.01 and final time t = 4π , and list them in Table 1 for point xj −1/3 and
in Table 2 for point xj . We see the two errors match very well with each other.
Note that when measuring as finite element solutions (using 40 uniformly spaced sampling points per cell), the scheme accuracy in both L2 and L∞ is 3rd order, see the left
column of Table 3.
Case II: β0 = 3 β1 =
1
12
The numerical flux now becomes
ux = 3
[u]
x
+ ūx +
[uxx ].
x
12
648
J Sci Comput (2012) 52:638–655
The Fourier analysis gives the DDG solution with such numerical flux as
2
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) +
5
2
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
648
(3.11)
and
49
2
x 4 e−α t sin(αxj ) + O(x 6 ).
(3.12)
77760
Again, the result for Im{uj + 1 (t)} is similar to Im{uj − 1 (t)}. Similar to Case I, 3rd order of
3
3
accuracy (the leading term in (3.11)) is obtained for points xj ± 1 and 4th order for point xj .
3
We also compare the numerical solution errors and the analytical errors with α = 0.01 and
t = 4π , and the two errors match well with each other. We use this example to show that
the scheme accuracy is not sensitive to the choice of β0 for the DDG method (2.1) with
quadratic approximations. In Table 3 we list the errors measured as a finite element method
for this case and case I together, and we see the numerical solution errors and orders are
similar to each other with different β0 ’s.
2
Im{uj (t)} = e−α t sin(αxj ) +
Case III: β0 = 3 and β1 =
1
4
For this case, the numerical flux is
ux = 3
[u]
x
+ ūx +
[uxx ],
x
4
(3.13)
and the Fourier analysis of the DDG method gives the solutions as
2
Im{uj − 1 (t)} = e−α t sin(αxj − 1 ) +
3
3
t
2
x 2 e−α t sin(αxj − 1 ) + O(x 3 ),
3
18
(3.14)
and
t
2
x 2 e−α t sin(αxj ) + O(x 4 ).
(3.15)
18
The result for Im{uj + 1 (t)} is similar to Im{uj − 1 (t)}. Notice the leading errors degenerate
3
3
to 2nd order for all the three points xj − 1 , xj , xj + 1 . We list the numerical errors and the
3
3
analytical errors for point xj − 1 in Table 4. Again the numerical solution errors and the
3
analytical errors match well on the three points. In Table 5 at the left column we list the
numerical errors for point xj and at the right column we list the errors and orders of the
2
Im{uj (t)} = e−α t sin(αxj ) +
Table 4 DDG scheme (2.1) with Case III numerical flux. L2 and L∞ errors for u, measured at the point
xj − 1 of the cells
3
x
Numerical solutions
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
1.23E-04
–
1.75E-04
–
1.07E-04
–
1.52E-04
–
2π/80
2.79E-05
2.14
3.95E-05
2.15
2.69E-05
2.00
3.80E-06
2.00
2π/160
6.78E-06
2.04
9.59E-06
2.04
6.71E-06
2.00
9.49E-06
2.00
2π/320
1.68E-06
2.01
2.38E-06
2.01
1.68E-06
2.00
2.37E-06
2.00
2π/640
4.20E-07
2.00
5.94E-07
2.00
4.20E-07
2.00
5.94E-07
2.00
Order
J Sci Comput (2012) 52:638–655
649
Table 5 DDG scheme (2.1) with Case III numerical flux
Numerical solutions at the point xj
h
Measuring as finite element solutions
L2 error
Order
L∞ error
Order
2π/40
1.08E-04
–
1.52E-04
2π/80
2.69E-05
2.00
3.80E-05
2π/160
6.72E-06
2.00
2π/320
1.68E-06
2.00
2π/640
4.20E-07
2.00
L∞ error
L2 error
Order
–
7.07E-04
–
1.78E-04
–
2.00
1.71E-04
2.05
3.97E-05
2.16
9.50E-06
2.00
4.25E-05
2.01
9.60E-06
2.05
2.37E-06
2.00
1.06E-05
2.00
2.38E-06
2.01
5.94E-07
2.00
2.65E-06
2.00
5.94E-07
2.00
Order
method measured as finite element solutions (using 40 uniformly spaced sampling points
per cell). Note that only 2nd order is obtained in L2 and L∞ . Here we take α = 0.1 and
t = 4π . This example shows that the scheme accuracy is sensitive to the choice of β1 for the
DDG method with quadratic approximations. With β1 = 1/12, the scheme degenerates to a
2nd order method. The numerical results match well with the theoretical predictions through
the Fourier analysis.
3.2 DDG Method with Interface Corrections
In this section, we perform Fourier analysis on the refined DDG method, namely the DDG
with interface corrections (2.3) with quadratic p 2 polynomial approximations. We study two
1
), to illustrate that there exists a large
cases with I: (β0 , β1 ) = (3, 14 ) and II: (β0 , β1 ) = (3, 12
class of admissible numerical fluxes leading to optimal convergence for the refined DDG
method with interface corrections. The following Fourier analysis confirms that 3rd order of
accuracy is obtained with such admissible numerical fluxes.
Case I: β0 = 3 and β1 =
1
4
The numerical flux is
[u]
x
+ ūx +
[uxx ].
x
4
With Fourier analysis the solution at point xj −1/3 is given as
ux = 3
2
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) +
23
2
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
648
(3.16)
and the solution at point xj is given as
2
Im{uj (t)} = e−α t sin(αxj ) −
19 + 72t
2
x 4 e−α t sin(αxj ) + O(x 6 ).
8640
(3.17)
Result for Im{uj + 1 (t)} is similar to Im{uj − 1 (t)}. We see the leading error for points xj ± 1
3
3
3
is 3rd order, and is 4th order for point xj . These analytical results are consistent with the
numerical results at the points, see Table 6 for point xj −1/3 and Table 7 for point xj with
α = 0.01 and t = 4π .
When measuring as finite element solutions, the scheme accuracy (in L2 and L∞ ) is 3rd
order, see the left column of Table 8.
Case II: β0 = 3 and β1 =
1
12
650
J Sci Comput (2012) 52:638–655
Table 6 Refined DDG scheme (2.3) with Case I numerical flux. L2 and L∞ errors for u, measured at the
point xj − 1 of the cells
3
Numerical solutions
x
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
6.83E-05
–
9.64E-05
–
9.72E-05
–
1.37E-04
–
2π/80
1.20E-05
2.50
1.70E-05
2.50
1.21E-05
3.00
1.72E-05
3.00
2π/160
1.52E-06
2.99
2.15E-06
2.99
1.52E-06
3.00
2.15E-06
3.00
2π/320
1.90E-07
3.00
2.68E-07
3.00
1.90E-07
3.00
2.68E-07
3.00
2π/640
2.37E-08
3.00
3.35E-08
3.00
2.37E-08
3.00
3.35E-08
3.00
Order
Table 7 Refined DDG scheme (2.3) with Case I numerical flux. L2 and L∞ errors for u, measured at the
center xj of the cells
Numerical solutions
x
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
1.06E-06
–
1.50E-06
–
9.50E-07
–
1.34E-06
–
2π/80
5.96E-08
4.15
8.43E-08
4.15
5.94E-08
4.00
8.40E-08
4.00
2π/160
3.71E-09
4.01
5.25E-09
4.01
3.71E-09
4.00
5.25E-09
4.00
2π/320
2.32E-10
4.00
3.30E-10
4.00
2.32E-10
4.00
3.30E-10
4.00
2π/640
1.45E-11
4.00
2.05E-11
4.00
1.45E-11
4.00
2.05E-11
4.00
Order
Table 8 Refined DDG scheme (2.3) solution errors and orders with numerical flux of Case I and Case II
Case I
x
Case II
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
3.13E-04
–
1.04E-04
–
6.56E-05
–
2.37E-05
–
2π/80
5.58E-05
2.49
2.00E-05
2.38
1.12E-05
2.56
3.85E-06
2.62
2π/160
7.04E-06
2.99
2.52E-06
2.99
1.41E-06
2.99
4.84E-07
2.99
2π/320
8.80E-07
3.00
3.15E-07
3.00
1.76E-07
3.00
6.05E-08
3.00
2π/640
1.10E-07
3.00
3.94E-08
3.00
2.20E-08
3.00
7.56E-09
3.00
Order
The numerical flux becomes
ux = 3
[u]
x
+ ūx +
[uxx ],
x
12
and the Fourier analysis gives the DDG interface correction solutions as
2
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) +
5
2
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
648
(3.18)
for point xj −1/3 , and
2
Im{uj (t)} = e−α t sin(αxj ) +
11 − 12t
2
x 4 e−α t sin(αxj ) + O(x 6 )
8640
(3.19)
J Sci Comput (2012) 52:638–655
651
for point xj . The result for Im{uj + 1 (t)} is similar to Im{uj − 1 (t)}. Again, we obtain 3rd
3
3
order for points xj ± 1 , and 4th order for point xj . They are consistent with the numerical
3
results on the three points.
When measuring as finite element solutions, the scheme accuracy (in L2 and L∞ ) is 3rd
order. In Table 8 we list the solution errors and orders with the left column corresponding to
the Case I numerical flux, and the right column corresponding to the Case II numerical flux.
We see 3rd order of accuracy is obtained with these numerical fluxes. The above Fourier
analysis confirms that there exists a large class of admissible numerical fluxes that lead to
optimal convergence for the DDG method with interface corrections.
3.3 Variations of DDG Method with Symmetric/Nonsymmetric Structure
In this section we perform Fourier analysis to the two new DG methods (2.6) with numerical
flux (2.7). The first case is about the symmetric version of the DDG method with σ = +1
in (2.6), and the second case is about the nonsymmetric version with σ = −1 in (2.6). The
Fourier analysis gives 3rd order of convergence for both two schemes with p 2 approximations, and the analytical results show some interesting connections between the two DG
methods.
Case I: A variation of the DDG method with symmetric structure
The scheme is defined as
ut vdx +
ux vx dx − (
ux )j + 1 vj−+ 1 + (
ux )j − 1 vj+− 1 + ([u]
vx )j + 1 + ([u]
vx )j − 1 = 0
Ij
2
Ij
2
2
2
2
2
(3.20)
with the numerical flux given as
ux =
[u]
x
+ ux +
x
[uxx ],
12
vx =
[v]
x
+ vx +
x
[vxx ].
12
(3.21)
The Fourier analysis gives the DG solutions at points xj −1/3 , xj , xj +1/3 as
⎧
2
2
⎪
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) + 25−18t
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
⎪
2160
⎪
⎨
2
⎪
⎪
⎪
⎩
Im{uj (t)} = e−α t sin(αxj ) +
Im{uj +1/3 (t)} = e
−α 2 t
2
25−54t
x 3 e−α t
6480
sin(αxj +1/3 ) −
cos(αxj ) + O(x 4 ),
25+54t
x
6480
3 −α 2 t
e
(3.22)
cos(αxj +1/3 ) + O(x 4 ).
The leading errors in the solutions (3.22) are all 3rd order. Again they match well with
the numerical results at the points, see Table 9 for point xj −1/3 , Table 10 for point xj and
Table 11 for point xj +1/3 with α = 0.1 and t = 4π .
When measuring as finite element solutions, we obtain 3rd order of accuracy in L2 and
∞
L , see the left column of Table 12. Even in [26] we prove the optimal convergence of the
symmetric DDG method, here the Fourier analysis further confirms the optimal 3rd order of
accuracy for quadratic approximations.
Case II: A DG method with nonsymmetric structure
With σ = −1 in (2.6), the scheme is defined as
ut vdx +
ux vx dx − (ux )j + 1 vj−+ 1 + (ux )j − 1 vj+− 1 − ([u]
vx )j + 1 − ([u]
vx )j − 1 = 0.
Ij
Ij
2
2
2
2
2
2
(3.23)
652
J Sci Comput (2012) 52:638–655
Table 9 DDG method with symmetric structure (3.20)–(3.21). L2 and L∞ errors for u, measured at the
point xj − 1 of the cells
3
Numerical solutions
x
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
2.55E-05
–
3.62E-05
–
2.54E-05
–
3.59E-05
–
2π/80
3.18E-06
3.00
4.50E-06
3.00
3.18E-06
3.00
4.50E-06
3.00
2π/160
3.98E-07
3.00
5.62E-07
3.00
3.98E-07
3.00
5.62E-07
3.00
2π/320
4.97E-08
3.00
7.03E-08
3.00
4.97E-08
3.00
7.03E-08
3.00
2π/640
6.21E-09
3.00
8.79E-09
3.00
6.21E-09
3.00
8.79E-09
3.00
Order
Table 10 DDG method with symmetric structure (3.20)–(3.21). L2 and L∞ errors for u, measured at the
point xj of the cells
Numerical solutions
x
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
6.78E-06
–
9.58E-06
–
6.79E-06
–
9.61E-06
–
2π/80
8.49E-07
3.00
1.20E-06
3.00
8.49E-07
3.00
1.20E-06
3.00
2π/160
1.06E-07
3.00
1.50E-07
3.00
1.06E-07
3.00
1.57E-07
3.00
2π/320
1.33E-08
3.00
1.88E-08
3.00
1.33E-08
3.00
1.88E-08
3.00
2π/640
1.66E-09
3.00
2.35E-09
3.00
1.66E-09
3.00
2.35E-09
3.00
Order
Table 11 DDG method with symmetric structure (3.20)–(3.21). L2 and L∞ errors for u, measured at the
point xj + 1 of the cells
3
x
Numerical solutions
Predicted by analysis
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
5.34E-06
–
7.53E-06
–
5.06E-06
–
7.15E-06
–
2π/80
6.42E-07
3.06
9.07E-07
3.05
6.33E-07
3.00
8.95E-07
3.00
2π/160
7.94E-08
3.02
1.12E-07
3.01
7.91E-08
3.00
1.12E-07
3.00
2π/320
9.91E-09
3.00
1.40E-08
3.00
9.89E-09
3.00
1.40E-08
3.00
2π/640
1.24E-09
3.00
1.75E-09
3.00
1.24E-09
3.00
1.75E-09
3.00
Order
Take β0u = β0v = 1 in (2.7), the numerical flux becomes
ux =
[u]
x
+ ux +
x
[uxx ],
12
vx =
[v]
x
+ vx +
x
[vxx ].
12
(3.24)
Plug in this numerical flux (3.24) into (3.23), we end up having a DG scheme with no penalty
term (the solution jump term), which is similar to the Baumann-Oden method [5] except now
we have extra second derivative jump terms included in the scheme definition. We choose
such a numerical flux to show that the optimal 3rd order convergence is recovered with our
nonsymmetric DG method.
J Sci Comput (2012) 52:638–655
653
Table 12 L2 and L∞ errors and orders for the symmetric DDG method (3.20)–(3.21) and the nonsymmetric
DG method (3.23)–(3.24)
Symmetric case
x
Nonsymmetric case
L2 error
Order
L∞ error
Order
L2 error
Order
L∞ error
2π/40
8.45E-05
–
3.71E-05
–
8.45E-05
–
3.71E-05
–
2π/80
1.05E-05
3.01
4.62E-06
3.00
1.05E-05
3.00
4.62E-06
3.00
2π/160
1.32E-06
3.00
5.78E-07
3.00
1.32E-06
3.00
5.78E-07
3.00
2π/320
1.64E-07
3.00
7.22E-08
3.00
1.64E-07
3.00
7.22E-08
3.00
2π/640
2.06E-08
3.00
9.02E-09
3.00
2.06E-08
3.00
9.02E-09
3.00
Order
The Fourier analysis gives the DG solutions (3.23)–(3.24) at the points as
⎧
2
2
⎪
x 3 e−α t cos(αxj −1/3 ) + O(x 4 ),
Im{uj −1/3 (t)} = e−α t sin(αxj −1/3 ) + 25+54t
⎪
6480
⎪
⎨
2
⎪
⎪
⎪
⎩
Im{uj (t)} = e−α t sin(αxj ) +
2
2
−25+54t
x 3 e−α t
2160
Im{uj +1/3 (t)} = e−α t sin(αxj +1/3 ) +
cos(αxj ) + O(x 4 ),
2 2
−25+18t
x 3 e−α α t
2160
cos(αxj +1/3 ) + O(x 4 ).
(3.25)
The leading errors in (3.25) are all 3rd order. Again the analytical errors match well with the
numerical errors at the three points.
When measuring as finite element solutions, 3rd order of accuracy (in L2 and L∞ ) is
obtained and the errors and orders are listed in Table 12 at the right column. We see the
numerical solution errors of the nonsymmetric DG method are similar to the solution errors
of the symmetric DDG method. When comparing at point values, it shows that the two errors
are the same at xj in the absolute value sense, and the error of the Case I symmetric DDG
method at point xj − 1 is identical to the error of the Case II nonsymmetric DG method at
3
point xj + 1 in the absolute value sense, and similarly the error at point xj + 1 of Case I is
3
3
same as the error at point xj − 1 of Case II.
3
4 Conclusion
In this paper, we carry out the Fourier type error analysis of the recent four discontinuous
Galerkin methods with p 2 quadratic approximations, namely the DDG method [16]; the
DDG method with interface corrections [17]; the DDG method with symmetric structure
[26]; and a DG method with nonsymmetric structure [30]. We choose Lagrange polynomials
as basis functions, thus the degree of freedom can be taken as the solution values on the
uniformly distributed mesh points. The discontinuous Galerkin method is then rewritten
in a finite difference format. Standard Fourier type technique is used to analyze the error
behavior of the four discontinuous Galerkin methods. It is verified that optimal 3rd order
of accuracy can be obtained for all the four discontinuous Galerkin methods with suitable
numerical fluxes. The theoretical predicted errors match well with the numerical results.
References
1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer.
Anal. 19(4), 742–760 (1982)
654
J Sci Comput (2012) 52:638–655
2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/2002)
3. Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput. 31, 45–59 (1977)
4. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)
5. Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)
6. Brenner, S.C., Owens, L., Sung, L.-Y.: A weakly over-penalized symmetric interior penalty method.
Electron. Trans. Numer. Anal. 30, 107–127 (2008)
7. Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2008)
8. Cheng, Y., Shu, C.-W.: Superconvergence and time evolution of discontinuous Galerkin finite element
solutions. J. Comput. Phys. 227, 9612–9627 (2008)
9. Cheng, Y., Shu, C.-W.: Superconvergence of local discontinuous Galerkin methods for one-dimensional
convection-diffusion equations. Comput. Struct. 87, 630–641 (2009)
10. Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E.: Advanced Numerical Approximation of Nonlinear
Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697. Springer, Berlin (1998). Papers from
the C.I.M.E. Summer School held in Cetraro, 23–28 June 1997. Edited by Alfio Quarteroni, Fondazione
C.I.M.E. [C.I.M.E. Foundation]
11. Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods. In:
Discontinuous Galerkin Methods, Newport, RI, 1999. Lect. Notes Comput. Sci. Eng., vol. 11, pp. 3–50.
Springer, Berlin (2000)
12. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convectiondiffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998) (electronic)
13. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated
problems. J. Sci. Comput. 16(3), 173–261 (2001)
14. Dawson, C., Sun, S., Wheeler, M.F.: Compatible algorithms for coupled flow and transport. Comput.
Methods Appl. Mech. Eng. 193(23–26), 2565–2580 (2004)
15. Gassner, G., Lörcher, F., Munz, C.D.: A contribution to the construction of diffusion fluxes for finite
volume and discontinuous Galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007)
16. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J.
Numer. Anal. 47(1), 475–698 (2009)
17. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 8(3), 541–564 (2010)
18. Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146(2), 491–519 (1998)
19. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical Report
LA-UR-73-479, Los Alamos Scientific Laboratory (1973)
20. Rivière, B., Wheeler, M.F., Girault, V.: A priori error estimates for finite element methods based on
discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39(3), 902–931 (2001)
(electronic)
21. Shu, C.-W.: Different formulations of the discontinuous Galerkin method for the viscous terms. In: Shi,
Z.-C., Mu, M., Xue, W., Zou, J. (eds.) Advances in Scientific Computing, pp. 144–155. Science Press,
Beijing (2001)
22. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes.
J. Comput. Phys. 77(2), 439–471 (1988)
23. Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes.
II. J. Comput. Phys. 83(1), 32–78 (1989)
24. van Leer, B., Nomura, S.: Discontinuous Galerkin for diffusion. In: Proceedings of 17th AIAA Computational Fluid Dynamics Conference, 6 June (2005). AIAA-2005-5108
25. Vidden, C., Liu, H., Yan, J.: Admissibility analysis for direct discontinuous Galerkin method and its
variations (2011, in preparation)
26. Vidden, C., Yan, J.: Direct discontinuous Galerkin method for diffusion problems with symmetric structure. SIAM J. Numer. Anal. (2011, under review)
27. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer.
Anal. 15, 152–161 (1978)
28. Tadmor, E., Liu, Y.-J., Shu, C.-W., Zhang, M.: L2 stability analysis of the central discontinuous Galerkin
method and a comparison between the central and regular discontinuous Galerkin methods. ESAIM:
Math. Model. Numer. Anal. 42, 593–607 (2008)
J Sci Comput (2012) 52:638–655
655
29. Tadmor, E., Liu, Y.-J., Shu, C.-W., Zhang, M.: Central local discontinuous Galerkin methods on overlapping cells for diffusion equations. ESAIM: Math. Model. Numer. Anal. 45, 1009–1032 (2011)
30. Yan, J.: A discontinuous Galerkin method for nonlinear diffusion problems with nonsymmetric structure
(2011, in preparation)
31. Zhang, M., Shu, C.-W.: An analysis of three different formulations of the discontinuous Galerkin method
for diffusion equations. Math. Models Methods Appl. Sci. 13(3), 395–413 (2003). Dedicated to Jim
Douglas, Jr. on the occasion of his 75th birthday
32. Zhang, M., Shu, C.-W.: An analysis of and a comparison between the discontinuous Galerkin and the
spectral finite volume methods. Comput. Fluids 23, 581–592 (2005)